Wiener-Hopf equations technique for variational inequalities

Noor, Muhammad Aslam

doi:10.1007/bf03012270

Cited by 13 publications

(12 citation statements)

References 64 publications

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“…which is exactly the general variational inequality (9). For formulation, numerical methods, sensitivity analysis, dynamical system and other aspects of general variational inequalities, see [109,110,122,144] and the references therein.…”

Section: Definition 16mentioning

confidence: 99%

“…From Lemma 3, it follows that the variational inequalities (9) and the Wiener-Hopf equations (26) are equivalent. This alternative equivalent formulation is used to suggest and analyze a wide class of efficient and robust iterative methods for solving general variational inequalities and related optimization problems, see [90,91,99,109] and the references therein.…”

Section: Lemmamentioning

confidence: 99%

“…which is called the equilibrium problem, that was introduced and studied by Blum and Oettli [19]. For the recent applications and development, see [102,109,153] and the references therein. (II).…”

Section: General Equilibrium Problemsmentioning

confidence: 99%

“…T u, η(g(v), g(u)) = T u, η(g(v), g(w)) + T u, η(g(w), g(u)) . (111) From (108), (109), (110) and 111, we obtain…”

Section: Theorem 13mentioning

confidence: 99%

“…Section 5 is devoted to the concept of projected dynamical system in the context of variational inequalities, which was introduced by Dupuis and Nagurney [35] by using the fixedpoint formulation of the variational inequalities. For the recent development and applications of the dynamical systems, see [13,34,35,40,41,56,75,79,109,115,116,122]. In this technique, we reformulate the variational inequality problem as an initial value problem.…”

Section: Introductionmentioning

confidence: 99%

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New Trends in General Variational Inequalities

Noor

Rassias

2020

Acta Appl Math

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It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, the auxiliary principle and the penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.

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Section: Definition 16mentioning

confidence: 99%

Section: Lemmamentioning

confidence: 99%

Section: General Equilibrium Problemsmentioning

confidence: 99%

“…T u, η(g(v), g(u)) = T u, η(g(v), g(w)) + T u, η(g(w), g(u)) . (111) From (108), (109), (110) and 111, we obtain…”

Section: Theorem 13mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New Trends in General Variational Inequalities

Noor

Rassias

2020

Acta Appl Math

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Three-Step Iterative Algorithms for Multivalued Quasi Variational Inclusions

Noor

2001

Journal of Mathematical Analysis and Applications

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In this paper, we suggest and analyze some new classes of three-step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique. New iterative algorithms include the Ishikawa, Mann, and Noor iterations for solving variational inclusions (inequalities) and optimization problems as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results.

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Solvability of Multivalued General Mixed Variational Inequalities

Noor¹

2001

Journal of Mathematical Analysis and Applications

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In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued general mixed variational inequalities. The convergence of the proposed method only requires the partially relaxed strong monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.

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Wiener-Hopf equations technique for variational inequalities

Cited by 13 publications

References 64 publications

New Trends in General Variational Inequalities

New Trends in General Variational Inequalities

Three-Step Iterative Algorithms for Multivalued Quasi Variational Inclusions

Solvability of Multivalued General Mixed Variational Inequalities

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